394 XV NORMED ALGEBRAS AND SPECTRAL THEORY
for each function u e *%C(K), there exists a unique continuous operator T(u) on E such
that
for all xt y in E. In order that u\~*T(u) should be a representation of ^C(K) on E, it
is necessary and sufficient that
(x | y)
for all x, y in E and all u e ^C(K). In the last relation, the condition u e
be replaced by u e ^C(K).
2. (a) Deduce from Problem 1 a new proof of the fact that a representation u i— » T(u) of
^C(K) in a Hilbert space E can be uniquely extended to a representation u f— > jT(«)
of ^c(K) on E, satisfying the relation (15.10.4.1). (To prove that T(uv) = T(u)T(v)
for H, z; e ^C(K), proceed in two steps : first assume that one of the two functions is
continuous and use the relation fcr(«> • x, y = ^X.TCJO . y when w is continuous; then prove
that this relation remains valid for all u e <^C(K).)
(b) If a continuous operator Ke J2?(E) commutes with T(u) for all w e ^C(K), then
it commutes with T(v) for all y e ^C(K).
(c) If E is separable, show that each operator T(u), where u e ^C(K), is the limit, with
respect to the weak topology on &(E) (Section 12.15, Problem 9) of a sequence
(T(vn)), where vne&c(l£); and conversely. (For the converse, use (15.10.6(ii)) and
(b) above.)
3. With the notation of Problem 1, show that T(«) • x = 0 for x e E and u e ^C(K) if
and only if u is //.*, ^-negligible for all y in a total subset of E. The mapping wh-*T(w)
of ^c(K) into JS?(E) is injective if and only if the union of the supports of the measures
jit,, y is dense in K.
4. With the notation of Problem 1 , suppose that E is separable; let (a/) be a dense sequence
in E, and put vtj = | p,ait aj\. Then there exists a positive measure v on K such that the
relation v(N) = 0 is equivalent to the relation " vu(N) = 0 for all i and j " (13.15.8).
All the functions in WC(K.) which are equivalent with respect to v have the same image
under T, and we may therefore regard T as an injective homomorphism of Lg(v) into
^(E). Prove that N«(w) g ll^(«)ll for all functions u e ^g(v) (consider first the case
where u is a step-function). Consequently T is a bicontinuous mapping of Lg(*>) onto
its image in
5. Suppose that, under the hypotheses of (15.10.8) there exist two sequences (En,/xn)
and (E£, /4) satisfying the conditions of (15.10.9).
(a) Show that if p,n = 0 for n ^ 2, then f4 =* 0 for n 2> 2. (In other words, a topologic-
ally cyclic representation does not admit a canonical decomposition satisfying (15.10.9)
as the sum of at least two topologically cyclic representations.) To prove this, observe
that if a is a totalizing vector for T, and a'n is its projection on E'n , then a'n ^ 0 if E'n ^ 0
(i.e., if [Mfn ^ 0). If afn =£ 0 for some n ^ 2, show that there would exist a sequence (gm)
of functions belonging to ^C(K), such that T(gm) • a( tends to 0 and T(gm) • a'n tends to
a'n, and show that this contradicts the fact that p,'H is a measure with base fc-i . /*„